MathLabs

Problem 6

A convex quadrilateral ABCDABCD satisfies AB⋅CD=BC⋅DAAB\cdot CD=BC\cdot DA. Point XX lies inside ABCDABCD so that ∠XAB=∠XCD\angle XAB=\angle XCD and ∠XBC=∠XDA\angle XBC=\angle XDA. Prove that ∠BXA+∠DXC=180∘\angle BXA+\angle DXC=180^\circ.
Step 7 of 8: Apply the quadrilateral isogonal criterion
Previous stepNext step
Y isogonal to X⟺∠BXA+∠DXC=180∘Y\text{ isogonal to }X\Longleftrightarrow\angle BXA+\angle DXC=180^\circ
Detailed analysis

For two interior points X,YX,Y in a convex quadrilateral, reflecting the four vertex rays across the corresponding angle bisectors shows the standard criterion Y isogonal to X⟺∠BXA+∠DXC=180∘Y\text{ isogonal to }X\Longleftrightarrow\angle BXA+\angle DXC=180^\circ. One direction follows by adding the four reflected directed angles; the reverse follows by reflecting successively at A,B,C,DA,B,C,D. Step 6 supplies the isogonal point YY, so the criterion yields the required equality.